Transform techniques have played an important role in signal processing for many years. The need for transform techniques is associated with the fact that the amount of generated data (an input signal) may be so great that it results in impractical storage, processing and communication requirements. In such cases, representations beyond the simple sampling and quantization are needed. In addition, many kinds of frequency domain processing are known. When the input data is not in the frequency domain, a transformation of the data is generally required to apply such processing.
Image compression addresses the problem of reducing the amount of data required to represent a digital image. Various communication techniques recently developed, such as Asymmetric Digital Subscriber Line (ADSL), deal with the transmission of a great amount of data to a subscriber premise. ADSL can substantially transform an existing public information network from one limited to voice, text and low resolution graphics, to a powerful, ubiquitous system capable to bringing multimedia (including full motion video) to every home.
In order to process an input signal (e.g., an image of an object scene) using conventional ADSL techniques, the input signal must first be received by a transducer and converted into an appropriate form. The choice of a particular transform in a given application depends on the amount of reconstruction error that can be tolerated and the computational resources available. Fourier transform is very popular because of its wide range of applications. Fourier transformations can be performed electronically using a suitable computer and software. For electronic processing, the input data presented in a time domain is converted into a frequency domain and vice versa, and for coherent optical processing the input data is converted into amplitude transmittance variations.
Processing by a computer is usually serial in nature and the processing speed is very limited. The use of an array processor increases the amount of parallelism, as well as the processing speed. True real time (speed of light) processing, however, is still not possible with this approach.
Coherent optical processors can perform the Fourier transformation in real time. However, spatially coherent illumination suffers from phenomenon such as a speckle effect, i.e., the appearance of bright dots (interference phenomena) in the output correlation plane. When dealing with pattern recognition, which is part of an image processing technique, this effect is generally undesirable, since speckle can cause a false alarm in identification of the object.
The use of incoherent light, avoids the speckle effect (reducing signal noise) and may increase the dynamic range of the resultant transform. Since optical processing is frequently performed over the intensity rather than the field amplitude, an incoherent system has a superior dynamic range over a coherent system. Incoherent light based systems are generally less sensitive to component deformation (e.g., flatness of a spatial light modulator), thus reducing the severity of component specifications, as compared to those of the coherent light based system.
Techniques aimed at performing the Fourier and related transforms utilizing spatially incoherent light have been developed. A shearing interferometer based technique appears to be an attractive technique of the kind specified. This technique is also useful for measuring wavefront parameters. The constructional and operational principles of the shearing interferometer are disclosed in the following publications:                (1) “OTF Measurements With a White Light Source: an Interferometric Technique”, J. C. Wyant, Applied Optics 14, 1619 (1975);        (2) “Use of an AC Heterodyne Lateral Shear Interferometer With Real-Time Wavefront Correction Systems”, J. C. Wyant, Applied Optics. 14, page 2622, (1975).        
A joint transform correlator (JTC), based on the shearing interferometer, is disclosed in the following publication:                (3) “Joint Transform Correlator With Incoherent Output”, D. Mendlovic et al., JOSA A11, 3201-3205 (1994).        
Generally speaking, the shearing interferometer based technique deals with input objects that are (quasi-) monochromatic, but spatially incoherent. In general, this means that points in the input signal are (locally) temporally coherent but spatially incoherent (with the other points). Herein, such objects are termed “locally temporally coherent”.
The principle of shearing interferometer based techniques can be thought of as the creation of an interference pattern in an output plane. This interference pattern can be formed by two light sources each corresponding to an input signal. For example, one of the sources may be a vector or an array of locally spatially coherent light sources. Each source is the inverse image (either real or virtual) of another source, and each position in each of the sources is coherent only with its image source. To this end, a common shearing interferometer optical setup is provided with two signals indicative of the object, wherein each of these two signals is obtained by imaging each position on the object (which is spatially coherent with itself), thus enabling the interference between these two signals.
FIGS. 1A and 1B illustrate known optical setups 1A and 1B of, respectively, a shearing interferometer and a spatially incoherent JTC utilizing the same. To facilitate understanding, the same reference numbers are used for identifying those components, which are common in the setups 1A and 1B.
The shearing interferometer setup 1A is composed of an incoherent light source assembly 2 composed of spatially continuous, spatially incoherent and locally temporally coherent light, a beam splitter 3, a regular mirror 4 that creates a virtual coherent image of each point source, and a corner prism 5. There are two optical paths: one with via regular mirror 4, and the other via corner prism 5. The path of the regular mirror 4 reflects the input spatially incoherent image as is, and the path with the corner prism 5 provides a reflected mirror image. Thus, the wavefronts emanating from both optical paths interfere. This interference can be detected in an output plane OP (located downstream of the beam splitter), where an output acquisition device, e.g., CCD, is placed (not shown). Interference based on source- and mirror-images is known in the art, e.g., the Lloyd's mirror (Born & Wolf, Principles of optics, 1980, p. 262).
In the incoherent JTC setup 1B, the shearing interferometer of FIG. 1A is associated with a conventional coherent system 6 for performing a Fourier transform. The system 6 comprises a coherent light source 26 producing an input object, a lens 7, and an optional filter (not shown). Mounted in the optical path of light ensuing from the system 6 and propagating towards the shearing interferometer 1A, is a rotating diffuser 9.
Referring additionally to FIG. 1A, due to the light passage through the beam splitter 3, teach point on the input image is doubled. The fact that each point in the input image is temporally coherent only with one single point in its mirrored image provides a separate interference pattern due to each point of the image. Due to the wave nature of light, the free space propagation of two coherent points is an interference pattern with a frequency proportional to the distance between these points. All other parts of the image are fully incoherent with the two points, thus the intensity follows the cosine transform. This is described in the above publication (3).
The mathematical analysis in (3) shows that the amplitude impulse response of the system is as follows:δ(x−x0)→cos(kx0ν)  (1)Here, δ is the delta functional; x0 is the shifted center of the input signal (information to be transmitted); k is a constant associated with the geometry of the shearing interferometer, e.g., k=2π/λz corresponds to the coordinate of the output plane.
Since for an impulse at the input, the output is purely coherent, the output function is bipolar (includes positive and negative values). However, available detectors are sensitive to intensities, rather than fields. Thus, the intensity of the impulse response must be considered:I(x−x0)→cos2(kx0ν)=0.5(1+cos(2kx0ν))  (2)
It should be noted that the sum of many incoherent signals obtained, for example, from many discrete point sources of the input object is represented by the sum of their intensities, rather than their fields.
The intensity also acts as a cosine transform, but with a certain bias. The mathematical analysis of the same are given in the above publication (3). Evidently, this optical setup needs no lens for performing the Cosine transform.